Geotechnical Engineering

DHARM

STABILITY OF EARTH SLOPES 323

From Eq. 9.8,

1 =

c

γββzcsin cos

or zc =

c

γββsin cos

...(Eq. 9.9)

Thus for a given value of β, zc is proportional to cohesion and inversely proportional to
the unit weight.

From Eq. 9.9.

c

γ.zc

= sin β cos β ...(Eq. 9.10)

c

γ.zc

F

HG

I

KJ

is a dimensionless quantity and is called the ‘stability number’, it is designated by Sn.

By combining Eqs. 9.8 and 9.10, we get F = zc/z ...(Eq. 9.11)
Thus, the factor of safety with respect to cohesion is the same that with respect to depth.
The stability number concept facilitates the preparation of charts and tables for slope stability
analysis in more complex situations, especially in the case of finite slopes to be dealt with
later.

9.2.3 Infinite Slope in Cohesive-Frictional Soil

Let us consider an infinite slope in a cohesive-frictional soil as shown in Fig. 9.6.

zc

z

Cohesive-

frictional

soil

Ledge

e

O s

Strength

envelope

T 1

f

f

b

T 2

T

R

P

(a) Infinite slope in cohesive-frictional

soil-critical depth

(b) Relation between strength envelope

and angle of slope

t

Fig. 9.6 Infinite slope in a cohesive-frictional soil
It is obvious that for a slope with an angle of inclination less than or equal to φ, the
shearing stress will be less than the shearing strength for any depth, as represented by the
line OP; the slope will be stable irrespective of the depth in that case. If the slope is inclined at
an angle β greater than φ, it cuts the strength envelope at some point such as R. The point R
represents the state of stress at a certain depth at which the shearing stress equals the shear-
ing strength and hence denotes incipient failure. For any depth less than that represented by
R, the shearing stress will be less than the shearing strength and hence the slope remains
stable.